How to find value of $k$ and $p$

Question

Im so confused How can i solve this problem

the given below is:

1.Find the $k$; $2k+1,3k+4,7k+6$ in geometric

2.Find the $k$; $k-3,k+2,k+3$ in geometric

3.Find the $p$; $p+7,3p+9,p+3$ in arithmetic

Thanks for the solution any answer is highly appreciated.

Answer 1

Hint: for three consecutive terms $a,b,c$ in a geometric progression, we must have $ac=b^2$, and if they are instead in an arithmetic progression we must have $a+c=2b$. Thus you can formulate a quadratic equation for the geometric problems and a linear one for the arithmetic problems.